Sufficient conditions for graphs with girth [formula omitted] to be maximally [formula omitted]-restricted edge connected

Abstract

For a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X]is connected}, where X̄=V∖X. G is maximally k-restricted edge connected (λk-optimal for short) if λk(G)=ξk(G). The k-restricted edge connectivity is more refined network reliability index than the edge connectivity. In this paper, for graph G with girth g≥2k+1 and minimum degree δ≥k, we present a lower bound on the cardinality of λk-atoms of G when λk(G)<ξk(G). Moreover, we give some sufficient conditions for graphs with girth g to be λk-optimal.